Question 949017
If A and B are independent events, P(A) = 0.35, and P(B) = 0.35, find the probabilities below. (Enter your answers to four decimal places.)
(a) P(A ᑎ B)
<pre>
Since they are independent events, by definition of independent
events, P(A &#5198; B) = P(A)×P(B) = 0.35×0.35 = 0.1225 
</pre>
(b) P(A &#5196; B)
<pre>
P(A &#5196; B) = P(A) + P(B) - P(A &#5198; B) = 0.35 + 0.35 - 0.1225 = 0.5775
</pre>
(c) P(A | B) 
<pre>
Since they are independent, knowing that B has occurred does not
affect the probability of A, so it's the same as P(A) = 0.35.
But you can do it by the formula for conditional probability, and
get the same thing:

P(A | B) = P(A &#5198; B)/P(B) = 0.1225/0.35 = 0.35 

(d) P(A<sup>c</sup> &#5196; B<sup>c</sup>) 

P(A<sup>c</sup>) = 1-P(A) = 1-0.35 = 0.65
P(B<sup>c</sup>) = 1-P(B) = 1-0.35 = 0.65

If two events are independent, so are their complements*, so
P(A<sup>c</sup> &#5198; B<sup>c</sup>) = P(A<sup>c</sup>)×P(B<sup>c</sup>) = (0.65)(0.65) = 0.4225

P(A<sup>c</sup> &#5196; B<sup>c</sup>) = P(A<sup>c</sup>) + P(B<sup>c</sup>) - P(A<sup>c</sup> &#5198; B<sup>c</sup>) = 0.65 + 0.65 - 0.4225 = 0.8775.

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*If you need to prove that,

P(A<sup>c</sup> &#5198; B<sup>c</sup>)              <-- use DeMorgan's law:
= P[(A &#5196; B)<sup>c</sup>] 
= 1 - P(A &#5196; B) 
= 1 - [P(A) + P(B) - P(A &#5198; B)] 
= 1 - [P(A) + P(B) - P(A)×P(B)]
= 1 - P(A) - P(B) + P(A)×P(B)    <--factor -P(B) out of last two terms 
= 1 - P(A) - P(B)×[1 - P(A)] 
= P(A<sup>c</sup>) - P(B)×P(A<sup>c</sup>)             <--factor P(A<sup>c</sup>) out of both terms 
= P(A<sup>c</sup>)×[1-P(B)] 
= P(A<sup>c</sup>)×P(B<sup>c</sup>) 

Edwin</pre>